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From the Fun Fact files, here is a Fun Fact at the Easy level:

Repunit Fun

A repunit is a whole number consisting of only 1's, such as 1, 111, or 111111111. (It's like a repeating unit.)

These numbers have some fun properties. For instance, many have already noted that the square of a repunit exhibits a nice pattern:

111*111 = 12321,

1111*1111 = 1234321,

11111*11111 = 123454321,


though this pattern is somewhat messed up by carrying digits when the repunit size is bigger than 9.

Here's another less well known pattern:

222+(333)2 = 111111,

2222+(3333)2 = 11111111,

22222+(33333)2 = 1111111111.


You might see if you can figure out why.

Presentation Suggestions:
Can you find other cool properties of repunits?

The Math Behind the Fact:
This property and many such properties can be proved by writing an n-digit repunit as:

(10n - 1 )/9

which follows from noting the repunit is a sum of a geometric series with ratio 10. For n=4, this is just the identity 1111 = 9999/9.

You might also check out the Fun Facts Multiplication by 11 and Multiplication by 111.

How to Cite this Page:
Su, Francis E., et al. "Repunit Fun." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Subjects:    number theory
Level:    Easy
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
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